Functiones et Approximatio Commentarii Mathematici

The dynamical Mordell-Lang problem for Noetherian spaces

Jason P. Bell, Dragos Ghioca, and Thomas J. Tucker

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Let $X$ be a Noetherian space, let $\Phi:X\longrightarrow X$ be a continuous function, let $Y\subseteq X$ be a closed set, and let $x\in X$. We show that the set $S:=\{n\in\mathbb{N}\colon \Phi^n(x)\in Y\}$ is a union of at most finitely many arithmetic progressions along with a set of Banach density zero. In particular, we obtain that given any quasi-projective variety $X$, any rational map $\Phi:X\longrightarrow X$, any subvariety $Y\subseteq X$, and any point $x\in X$ whose orbit under $\Phi$ is in the domain of definition for $\Phi$, the set $S$ is a finite union of arithmetic progressions together with a set of Banach density zero. This answers a question posed by Laurent Denis [7]. We prove a similar result for the backward orbit of a point and provide some quantitative bounds.

Article information

Funct. Approx. Comment. Math., Volume 53, Number 2 (2015), 313-328.

First available in Project Euclid: 17 December 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37P55: Arithmetic dynamics on general algebraic varieties
Secondary: 11G99: None of the above, but in this section

dynamical Mordell-Lang conjecture Noetherian space Banach density


Bell, Jason P.; Ghioca, Dragos; Tucker, Thomas J. The dynamical Mordell-Lang problem for Noetherian spaces. Funct. Approx. Comment. Math. 53 (2015), no. 2, 313--328. doi:10.7169/facm/2015.53.2.7.

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